Relativistic spin-momentum locking in ferromagnets

The relativistic spin-momentum locking has been proven in time-reversal-breaking classes of materials with zero net magnetization in the non-relativistic limit, such as altermagnets and other non-collinear magnets. However, what happens in the relativistic case to the spin-momentum locking in a ferromagnetic environment has not yet been investigated. Using density functional theory calculations on prototypical ferromagnets such as orthorhombic SrRuO₃, hexagonal CrTe and CrAs with the NiAs crystal structure, half-Heusler MnPtSb, and fcc Ni, we aim to show the presence of relativistic spin-momentum locking in different classes of ferromagnets and to unveil their properties. In SrRuO₃, the antisymmetric exchange interaction produces a spin canting orthogonal to the easy axis, but even when the canted magnetic moment in real space is forbidden, relativistic spin-momentum locking shows sizable contributions in k-space. Subdominant components of centrosymmetric ferromagnetic materials with magnetic sites connected by rotational symmetry host a single spin-momentum locking similar to altermagnets, while noncentrosymmetric MnPtSb hosts interplay with the relativistic p-wave, and fcc Ni shows a more complex behavior with a combination of two spin-momentum locking patterns. Because ferromagnets typically have larger bandwidths than altermagnets, they provide a promising platform for observing even-wave relativistic spin-momentum locking and associated emergent phenomena. From an application standpoint, relativistic spin-momentum locking governs symmetry-allowed spin Hall currents, spin photocurrents, the Rashba-Edelstein effect and other momentum-dependent spin responses in k-space.